GNOMON

[kram1032]

It's still very unclear to me what you are talking about.
Are you saying that fractals always are recursions while gnomons are more generic maps of similar underlying simplicity?

And what exactly are you showing in those pictures? What kinds of transforms are you talking about?

[Chillheimer]

I have to disappoint you. I still don't get it, although I'm really trying.. 

First I´ll try to define what I mean by fractal
   I think this is a very simple process, since nature always uses simple methods.

the process I'd call iteration of the formula.

  In this case, recursion, but to start a process, you must have something to process.

these are the numbers that you use to fill the variables of the formula.

you have a formula, start with a chosen number and iterate the formula to achive a result...

I don't understand whats new or different for gnomons or where the connection is and what they are 'needed' for..


obviously I'm not the only one having a problem understanding what your point is..
Is there an official definition of "gnomon", that describes exactly what you are talking about? the definitions I find don't really seem to fit. can you recommend a link/read to better understand it?
Or is it a new concept that you are thinking of and use the term gnomon in your own way?

[stereoman]

The eye has a fairly simple gnomonic structure, and allows to understand the concept more clearly.
The lens is a special case, while forms, gets its feed through an artery, when it reaches its final state, the artery atrophy and disintegrates, the lens is formed by very thin wholly transparent layers, something like a squashed onion.
   Anyway, the curves of this lens are part of two overlapping and oppossed circles with different radius,thus this leads to further levels that are far beyond of our present aim.

[stereoman]

It's still very unclear to me what you are talking about.
Are you saying that fractals always are recursions while gnomons are more generic maps of similar underlying simplicity?

And what exactly are you showing in those pictures? What kinds of transforms are you talking about?

Let me put it this way, suppose you can edit de M-set, or better, the A-box.
Suppose a 3D program where you can model anything using as a base a flexible sphere, this exists, Mudbox, or Zbrush.
 Now, apply recursive formulas to it, is so simple.
 The gnomon applies transforms trough time, this is the point, and here is where is the difference, because trough gnomons you can mimic any natural growth.
  The gnomon includes some nested layers wich develop trough time, each carrying its own info.

[Chillheimer]

then we probably just have different standpoints.

The gnomon applies transforms trough time, this is the point, and here is where is the difference, because trough gnomons you can mimic any natural growth.

I believe that fractals do exactly this perfectly.
I believe the difference between a pure mathematical fractal and nature is that these natural fractal forms react to influences from outside. this is why they don't look as perfect. a tree for example shows fractal branching. but the system is so complex, you have to think if wind direction, shade of other trees that will change its appearance to a more ragged, natural way.
I believe that we can explain this with 'multifractal systems'.
The tree (system 1) reacts to the weather (system 2), to other trees position-->shade (system 3) and so on..

Also, I see evolution, the tree of life as a grand fractal. where over time the formula changes and grows in a fractal, self similar way.
which is why it is soo complex after billions of years that we just can't (yet) reverse engineer the formula..

thats my view..

but I'll keep reading this thread, as different approaches to a topic don't have to contradict..

cheers!

[kram1032]

Ah, ANY transformation?
So what you are saying is that, unlike fractals where you tend to apply the same limited set of transformations over and over again, in a gnomon you could at any point in time switch to any other transformation? Is that the difference?

If that's the case, I'd think that gnomons really are just recursions, as found in fractals, in which you extend the parameter space of a single transformation to infinitely many dimensions.

And the exact way in which a transformation happens is based on all those parameters. And the choice of parameters can be partially random, partially fixed and partially determined by an arbitrary feedback loop.
(roughly corresponding to Noise/Errors, genetic adaption and learned adaption respectively in a living organism)

Is that about right?

[stereoman]

I have to disappoint you. I still don't get it, although I'm really trying.. 

the process I'd call iteration of the formula.
these are the numbers that you use to fill the variables of the formula.

you have a formula, start with a chosen number and iterate the formula to achive a result...

I don't understand whats new or different for gnomons or where the connection is and what they are 'needed' for..


obviously I'm not the only one having a problem understanding what your point is..
Is there an official definition of "gnomon", that describes exactly what you are talking about? the definitions I find don't really seem to fit. can you recommend a link/read to better understand it?
Or is it a new concept that you are thinking of and use the term gnomon in your own way?

 This thing does not exists yet, so has no name, gnomonic it´s the only name that fits with it´s purpose because of their expected behaviour.
 It´s   something that mimics life trough electronic pulses. sinchronized trough three levels, a kind of very sophisticated pencil, if you ask me, but this is too far for now.
   I need this tool, let me say it this way, I´m now planning some scenes, trough standard programs I can achieve a decent integration between  3D objects and fractal backgrounds, but there is no way to generate a fractal human head, whatever this be.
    But I have in mind a wonderful head made by small and countless iterations of the same head with all its details, hairdo, etc, moreover, I want different hues and shades for them, moreover, its outer cover is transparent and let see some organic inner structures made of different materials.
    It´s nearly imposible to achieve such an image with standard programs, or if it can be done it will cost a lot of time and effort, but this would be nothing for this tool .
    In your terms, it´s a kind of universal and editable formula, if I can say it.

[kram1032]

I am with Chillheimer on this one. It's really just a vastly expanded rule-set in interplay.
If you will, the fractals we are typically rendering here are just what theoretical physicists or biologists would probably call "toy systems" - very simple cases of much more general possible forms.
They have little direct application in the real world, as they are too simple / make too many assumptions, but they are very helpful to highlight specific intricate details about these kinds of structures in general.

Though even if you throw in more and more of the rules that would actually be in interplay in real life, it's still a highly self-recursive, iterated feedback-loop that ultimately very often ends up defining a fractal of some sort (a strange attractor, usually)

[stereoman]

Ah, ANY transformation?
So what you are saying is that, unlike fractals where you tend to apply the same limited set of transformations over and over again, in a gnomon you could at any point in time switch to any other transformation? Is that the difference?

If that's the case, I'd think that gnomons really are just recursions, as found in fractals, in which you extend the parameter space of a single transformation to infinitely many dimensions.

And the exact way in which a transformation happens is based on all those parameters. And the choice of parameters can be partially random, partially fixed and partially determined by an arbitrary feedback loop.
(roughly corresponding to Noise/Errors, genetic adaption and learned adaption respectively in a living organism)

Is that about right?

You get it I think

[stereoman]

then we probably just have different standpoints.
I believe that fractals do exactly this perfectly.
I believe the difference between a pure mathematical fractal and nature is that these natural fractal forms react to influences from outside. this is why they don't look as perfect. a tree for example shows fractal branching. but the system is so complex, you have to think if wind direction, shade of other trees that will change its appearance to a more ragged, natural way.
I believe that we can explain this with 'multifractal systems'.
The tree (system 1) reacts to the weather (system 2), to other trees position-->shade (system 3) and so on..

Also, I see evolution, the tree of life as a grand fractal. where over time the formula changes and grows in a fractal, self similar way.
which is why it is soo complex after billions of years that we just can't (yet) reverse engineer the formula..

thats my view..

but I'll keep reading this thread, as different approaches to a topic don't have to contradict..

cheers!

All viewpoinst are welcome so we can discuss.
 But, to me, any tree has a function that not depends on any evolution, and is the function what defines the shapes and sizes of the tree, in the same way, is the function what defines the shape and size of everything.
     So we try to go as near of the core function as posible .
    I agree that  fractals show some kind of growing, but I´ve never seen a decent fractal generated tree, nor a fractal generated terrain that can be used together with 3D models without further edit.
    

[stereoman]

I am with Chillheimer on this one. It's really just a vastly expanded rule-set in interplay.
If you will, the fractals we are typically rendering here are just what theoretical physicists or biologists would probably call "toy systems" - very simple cases of much more general possible forms.
They have little direct application in the real world, as they are too simple / make too many assumptions, but they are very helpful to highlight specific intricate details about these kinds of structures in general.

Though even if you throw in more and more of the rules that would actually be in interplay in real life, it's still a highly self-recursive, iterated feedback-loop that ultimately very often ends up defining a fractal of some sort (a strange attractor, usually)

 No problema with this, since , as above, so below.
  The interesting part for us is the level we can reach and work with.
  I agree,our present fractals are toy systems if compared with natural true systems ( and are impressive), this is the best we can approach them, and that´s why I look for a new and more powerful tool.

[kram1032]

Really, the amazing complexity of natural systems partially stems from the effective starting conditions and effective rules not only having interesting attractors but also from being dynamically changed in a fractal manner.

Note that I said "effective". - if you were to follow each individual particle and all the forces that play between all those particles exactly, you'd end up with a very simple (relatively speaking) rule set that underlies it all.
It's just that the sheer number of particles we are typically dealing with makes such a detail view beyond impractical. We have to coarse-grain and approximate the globally valid rules with local ones that are more circumstantial, hide more of the details but yield the right results (within the tolerances we allow through the approximations we choose) more directly. This abstracted, local rule-set is what actually changes over time. And since it is a zoomed-out version of the underlying much simpler but much less practical rules below, this local rule-set effectively is a fractal in itself.

This way, nature as naively defined by our own perception, is a fractal of fractals.

[kram1032]

I should clarify: The base-rules are "simpler" in that they consist of primitive notions (individual particles, only the four fundamental forces of physics - at least under our current description). They, however, end up giving much more complicated behavior, because you do not approximate away corner cases that give interesting scenarios that, however, are ultimately really hard to deal with.

This can especially be seen in thermodynamics where it's fairly straight forward to describe a gas of homogeneous, constant temperature, pressure, volume, particle number or chemical potential. But once you introduce spacial or temporal inhomogeneous regions or allow phase transitions, as found in so called non-equilibrium-thermodynamics, you very soon hit the very edge of what can currently be described. Our models just aren't good enough for that for now.

[stereoman]

I should clarify: The base-rules are "simpler" in that they consist of primitive notions (individual particles, only the four fundamental forces of physics - at least under our current description). They, however, end up giving much more complicated behavior, because you do not approximate away corner cases that give interesting scenarios that, however, are ultimately really hard to deal with.

This can especially be seen in thermodynamics where it's fairly straight forward to describe a gas of homogeneous, constant temperature, pressure, volume, particle number or chemical potential. But once you introduce spacial or temporal inhomogeneous regions or allow phase transitions, as found in so called non-equilibrium-thermodynamics, you very soon hit the very edge of what can currently be described. Our models just aren't good enough for that for now.

So, in your view, it´s not  posible to do it?
I assume it´s very difficult to implement  the full concept, but that´s why I think it must be done with small steps starting with some simplified 2D model, Is that a right approach?

[kram1032]

So, in your view, it´s not  posible to do it?

Depends on what precisely "it" is.
Also, I do not think any simulation would be impossible. Just... hard.
The more details you want, the harder and slower it gets.
More correctly, "it" might be impractical given our current understanding of the basic rules as well as our technological capabilities. It might become very feasible one day.
And yeah, step-by-step better approximation is a good way to get there and it's what scientists have been doing since at the very least the invention of Newton's / Leibniz' calculus.